Vortex pairs and dipoles on closed surfaces. (English) Zbl 1494.76023
Summary: We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller [Fields Inst. Commun. 73, 185–237 (2015; Zbl 1402.76034)] by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.
MSC:
| 76B47 | Vortex flows for incompressible inviscid fluids |
| 76M40 | Complex variables methods applied to problems in fluid mechanics |
| 30F99 | Riemann surfaces |
| 53Z05 | Applications of differential geometry to physics |
Keywords:
vortex pair; vortex dipole; Riemannian surface; geodesic curve; affine connection; projective connection; Robin function; Hamiltonian function; symplectic form; Green functionCitations:
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